Method of reducing polarization fluctuation inducing drift in resonator fiber optic gyro

ABSTRACT

There is provided a method of generating no error in the output of a resonator fiber optic gyro even when the polarization dependency loss is present in a ring resonator of the resonator fiber optic gyro. The relationship between ΔL and Δβ is set to satisfy the formula ΔβΔL=π+ 2 nπ [radian] (n: integer), or ΔβΔL is close to the value obtained from the formula, where ΔL is defined as the difference between the lengths L 1  and L 2  of two portions of a waveguide divided by a coupler and a polarization-rotating point in the ring resonator, and Δβ is defined as the difference in propagation constant of two axes of polarization having the waveguide. ΔL is set so that the error induced by the polarization fluctuation is minimized. Even when the polarization dependency loss is present in the coupler or the like, the ring resonator itself generates no errors thereby, and the error of the gyro output induced by the polarization fluctuation can be minimized in the resonator fiber optic gyro.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention relates to a sensor using a ring resonator,and more particularly, it relates to a method of reducing thepolarization fluctuation inducing drift in a resonator fiber optic gyroto measure the differential resonant frequency generated by the rotationbetween two light waves facing each other propagating in the ringresonator.

[0003] 2. Description of the Related Art

[0004] Fiber optic gyros (FOG: Fiber Optic Gyro) to detect therotational speed of an object for measurement making use of the SagnacEffect generated by the rotation include a resonator fiber optic gyro(R-FOG: Resonator Fiber Optic Gyro). The R-FOG can obtain the highsensitivity by the short fiber length making use of the sharp resonancecharacteristic of the ring resonator.

[0005] The ring resonator will be described below.

[0006] The R-FOG uses a reflector ring resonator or a transmitter ringresonator comprising an optical fiber and a coupler as a sensing loop.The reflector ring resonator comprises a sensing loop 33 and a coupler32 as shown in FIG. 9(a). The resonance characteristic shown in FIG.10(a) can be obtained when the laser beam is incident from a port 1 andthe intensity of the emitted light is observed at a port 2 to acquirethe characteristic of the incident light to the frequency. Thetransmitter ring resonator comprises a sensing loop 36 and two couplers35 and 37 as shown in FIG. 9(b). The resonance characteristic shown inFIG. 1OB can be obtained when the laser beam is incident from the port 1and the intensity of the emitted light is observed at the port 2. Thespace of the resonance points is referred to as the free spectrum range,and given as follows: $\begin{matrix}{\nu_{FSR} = \frac{c}{nL}} & (1)\end{matrix}$

[0007] where c is the velocity of light, n is the refractive index ofthe optical fiber, and L is the sensing loop length.

[0008] The fineness is the parameter showing the sharpness of theresonance, and defined by the formula (2). $\begin{matrix}{F = {\frac{\nu_{FSR}}{\Delta\nu} = \frac{\pi \sqrt{\alpha \quad R}}{1 - {\alpha \quad R}}}} & (2)\end{matrix}$

[0009] where Δv is the full width at half maximum of the resonancecharacteristic shown in FIG. 10, R is the branching ratio of thecoupler, and α is the loss in the ring resonator. Generally speaking,the detection sensitivity as the gyro is increased as the fineness isincreased.

[0010] The resonance characteristic can also be obtained similarly bysetting the X-axis as the phase difference between two light wavesdifferent by one trip propagating around the ring resonator as shown inFIG. 11. The space of the resonance points adjacent to each other isjust 2π.

[0011] The principle of detecting the rotation of the R-FOG is that,when the ring resonator is rotated at the angular velocity Ω, an opticalpath difference is generated due to Sagnac Effect in the optical pathlength of the clockwise light wave (CW light) and the counterclockwiselight wave (CCW light), and the difference is generated thereby in theresonant frequency of the CW light and the CCW light. The difference inresonant frequency is expressed as follows: $\begin{matrix}{{\Delta\nu}_{s} = {\frac{4S}{\lambda \quad L}\Omega}} & (3)\end{matrix}$

[0012] where S is the area surrounded by the ring resonator, and λ isthe wavelength of the oscillated laser beam. The angular velocity Ω isobtained by measuring the difference in the resonant frequency.

[0013]FIGS. 12 and 13 show the general configuration of the R-FOGs usingthe reflector ring resonator and the transmitter ring resonator,respectively.

[0014] The light emitted from laser beam sources 41 and 49 is branchedinto two by a beam splitter BS. The two branched lights pass throughlenses L_(1 and L) ₂, respectively, and are guided to the optical fiber,and incident in a coupler C1. The lights incident in ring resonators 46and 54 by the coupler C1 propagate in the loop clockwise (CW) andcounterclockwise (CCW).

[0015] The resonance characteristic is observed by a light receiver D1for the CW light and by a light receiver D2 for the CCW light,respectively. In order to detect the resonance point, both the CW lightand the CCW light are bias-modulated by the sinusoidal wave having thefrequency f_(n) and the frequency f_(m) by phase modulators PM2 and PM1before the lights are incident in the ring resonators 46 and 54. Thefrequency f_(n) and the frequency f_(m) are generated by oscillators (1)44 and 53 and oscillators (2) 45 and 52. The differential resonancecharacteristic can be obtained through the synchronous detection at thefrequency of the sinusoidal wave. The resonance point is the frequencyof the light wave at which the differential resonance characteristic iszero, and can be detected and traced through the feedback thereto.

[0016] In the R-FOG, the reciprocal effect to the CW light and the CCWlight is separated from the non-reciprocal effect due to Sagnac Effect.

[0017] Generally, for the reciprocal effect, a method for feedback tothe frequency of oscillation of the laser beam source based on theoutput of the light receiver of either the CW light or the CCW light isemployed in order to follow the resonance point to be shifted.

[0018] The method for bias modulation to detect the resonance point ispossible by implementing the binary frequency shift by the “digitalserrodyne” at a predetermined switching frequency as introduced in KazuoHotate and Michiko Harumoto, “RESONATOR FIBER OPTIC GYRO USING DIGITALSERRODYNE MODULATION”, IEEE J. Lightwave Technol., Vol. 15, No. 3, pp.466-473, 1997 (Literature 1).

[0019] In FIGS. 12 and 13, the CW light and the CCW light aresynchronously detected by a synchronous detection circuit LIA1 and asynchronous detection circuit LIA2, respectively, and the deviation inthe resonant frequency can be detected. The output of the synchronousdetection circuit LIA1 is inputted in laser frequency control circuits39 and 47, and control device so that the laser beam frequency f₀ agreeswith one resonance point with reference to the CW light. The output ofthe synchronous detection circuit LIA2 is obtained from the deviation inthe resonance point between the CW light and the CCW light as thevoltage value. This is the open loop output of the gyro.

[0020] In order to expand the range of detection, a closed loop systemusing the serrodyne (sawtooth) wave is used. The serrodyne wave works tochange the frequency of the light wave by the frequency of the serrodynewave with the amplitude thereof as the voltage value to give the phaserotation of 2π. An electro-optic modulator formed on a lithium niobate(LiNbO₃:LN) waveguide is extensive in frequency range, and used inmodulation by the serrodyne wave.

[0021] The output of the synchronous detection circuit LIA2 is inputtedin integrators 43 and 51 for integration control, and the output of theintegrators is inputted in a voltage control oscillator VCO to changethe frequency f₂ of the serrodyne wave generated by a serrodyne wavegenerator. The serrodyne wave is inputted in a waveguide type phasemodulator LN2, and controlled so that the CCW light agrees with theresonance point. The frequency of the CCW light becomes f₀+f₂. Theclosed loop output of the gyro can be obtained by counting the frequencyf₂ of the serrodyne wave.

[0022] In order to further improve the resolution, the serrodyne wavehaving a fixed frequency f₁ is inputted in the CW light using awaveguide type phase modulator LN1 simultaneously with the input of theserrodyne wave in the CCW light. The frequency of the CW light in thiscondition becomes f₀+f₁. As a result, the difference in the frequencyf₂-f₁ between the CCW light and the CW light becomes the closed loopoutput of the gyro.

[0023] Polarization fluctuation which is one of the error factors of theR-FOG will be described below.

[0024] The polarization fluctuation means the change of the polarizationcondition of the light wave by the unevenness of the waveguide or thepolarization dependency, and is considerably affected by theenvironmental conditions including the temperature or the like.

[0025] Thus, a polarization maintaining fiber (PMF: PolarizationMaintaining Fiber) is generally used. The polarization maintaining fiberhas two axes of polarization along which the linearly polarized wave canbe propagated.

[0026] However, even when the polarization maintaining fiber is used, itis practically impossible to selectively use one axis of polarizationdue to the crosstalk of the optical fiber (the waveguide) and thecoupler, and the assembly errors in manufacturing the ring resonator,and the resonance characteristic of the ring resonator has thecharacteristic of superposing two eigenstates of polarization (ESOP:Eigenstate of Polarization). The Eigenstate of Polarization (ESOP) meansthe polarization condition that the light wave is not changed when thelight wave makes one trip around the ring resonator, and corresponds totwo eigenvectors of the transfer matrix expressing one trip of the ringresonator.

[0027] Next, the condition that two ESOPs affect the gyro output will bedescribed below.

[0028] More correctly, in the ring resonator shown in FIG. 9, at leastone splice (fusion) point is required in the sensing loop as shown inFIG. 14. The fusion angle θ must be adjusted here so that the axes ofpolarization of fibers for fusion must agree with each other; however,the rotation of the axes of polarization is generated due to the angulardeviation. In addition, the crosstalk at the sensing loop and thepolarization coupling at the coupler or the lead part are generated.Thus, even when the laser beam of the linearly polarized wave isincident only on X-axis, the light wave is actually coupled with Y-axisin an unwanted manner, appearing an aspect different from an ideal one.

[0029] In the resonance characteristic observed by the light receiver,the resonance points corresponding to two ESOPs appear as shown in FIG.15. When ESOP having the resonance point used to detect the rotation isdefined as ESOP1, ESOP2 has an unwanted resonance point. The positionalrelationship of the resonance points of ESOP1 and ESOP2 is dependent onthe length of the resonator, i.e., the length of the sensing loop, andfluctuated according to the environmental temperature, etc. Theresonance point of ESOP2 is increased as the distance from ESOP1 isdecreased. FIG. 15 shows this condition in a part expressed by chainlines.

[0030] Approach of the resonance point of ESOP2 to the resonance pointof ESOP1, and duplication thereof lead to a very large error factor ofthe gyro output, which is quantitatively verified in K. Takiguchi and K.Hotate, “BIAS OF AN OPTICAL PASSIVE RING-RESONATOR GYRO CAUSED BY THEMISALIGNMENT OF THE POLARIZATION AXIS IN THE POLARIZATION-MAINTAININGFIBER RESONATOR”, IEEE J. Lightwave Technol., Vol. 10, No. 4, pp.514-522, 1992 (Literature 2). According to the literature, the resonancepoints of two ESOPs are duplicated when ΔβL which is the product of thedifference AP in the propagation constant of two axes of polarization ofthe polarization maintaining fiber used in the ring resonator and thelength L of the sensing loop fiber is ΔβL=2 mπ (m: integer), and theresonance points of two ESOPs are separated most from each other whenΔβL=π+2mπ (m: integer), and the unwanted resonance points of ESOP2 canbe minimized.

[0031] However, ΔβL is changed by at least π in the temperature changeof about 1° C., and in reality, duplication of the resonance pointscannot be avoided. In order to solve this problem, the fusion isimplemented with the axes of polarization of the polarizationmaintaining fiber twisted by 90° at the splice point in the ringresonator as shown in FIG. 16 (θ=90°), and this method has beendisclosed in U.S. Pat. No. 5,018,857, Sanders et al., “PASSIVE RESONATORGYRO WITH POLARIZATION ROTATING RING PATH” (Literature 3) and Sanders etal., “NOVEL POLARIZATION-ROTATING FIBER RESONATOR FOR ROTATION SENSINGAPPLICATIONS”, Proc. SPIE, Fiber Optic and Laser Sensors VII., Vol.1169, pp. 373-381, 1989 (Literature 4).

[0032] The rotation of 90° of the polarized wave in the ring resonatorequally excites two ESOPs as shown in FIG. 17, and since the resonancepoint of one ESOP is located at the center of the other resonance pointwhich repeatedly appears, and the resonance points of the two ESOPs donot duplicate even when the length of the resonator is changed.

[0033] In addition, if the angular deviation at the 90° splice point is1° (θ=89° or θ=91°), and the fineness of the resonator is not less than100, it has been analyzed that the error is not more than 10⁻⁷(radian/s) which is the accuracy required for the inertial navigation ofan aircraft in K. Takiguchi and K. Hotate, “EVALUATION OF THE OUTPUTERROR IN AN OPTICAL PASSIVE RING-RESONATOR”, IEEE Photon. Technol., Vol.3, No. 1, pp. 80-90, 1991 (Literature 5). However, it is assumed in thisanalysis that no polarization dependency is present in the ringresonator.

[0034] However, the experimental result that the polarization dependencywhich has been assumed to be absent works as a large error factor, andthe performance expected from the theoretical analysis cannot beachieved, is demonstrated in L. K. Strandjord and G. A. Sanders,“RESONATOR FIBER OPTIC GYRO EMPLOYING A PORARIZATION-ROTATINGRESONATOR”, Proc. SPIE, Vol. 1585, Fiber Optic Gyros: 15^(th)Anniversary Conference, 1991 (Literature 6).

[0035] In order to clarify the problems, the analysis shown inLiterature 5 will be described. This analytical method is theoreticallydeveloped in detail by Literature 2, and the models used in the analysisare substantially equivalent to each other. The main difference is thatthe angle θ at the splice point is set to be θ≈0° in the latteranalysis, while θ is set to be θ≈90° in the former analysis.

[0036]FIG. 18 shows a model of the ring resonator used in theseanalyses. P₁ and P₂ are polarizers connected to the lead part of thering resonator. However, in the analysis of the 90° splice (Literature5), no polarizer is inserted in the lead part of the ring resonator. Thecoupler is assumed to be free from any polarization dependency in bothanalyses. E_(0CW) and E_(0CCW) in FIG. 18 are the laser beam incident inthe ring resonator, where the incident light is assumed to be thelinearly polarized wave, and θ_(iCW) and θ_(iCCW) express the angulardeviation to the axes of polarization of the fiber. L means the lengthof the fiber of the sensing loop, and L₁ and L₂ mean the lengths of twoportions divided from the coupler 55 at the splice point. ΔL is thedifference therebetween.

[0037] According to Literature 2, the power, i.e., the resonancecharacteristic of one light to be observed by the light receiver issummarized in the following form:

|E _(dCW) /E _(0CW)|² =K ₁ |U ₁|² +K ₂ |U ₂|² +K ₃  (4)

[0038] where${U_{j}}^{2} = {K_{4}\left\lbrack {1 - \frac{K_{5}}{\left( {1 - K_{6}^{1/2}} \right)^{2} + {4K_{6}^{1/2}{\sin^{2}\left( {\beta_{j}{L/2}} \right)}}}} \right\rbrack}$

[0039] The resonance characteristic for the CCW light is similarthereto.

[0040] In the formula, K_(i) (i=1 to 6) is the constant determined bythe parameters of the ring resonator, and β_(j) (j=1, 2) is thepropagation constant of each ESOP. First, second and third terms in theformula (4) mean the interference components expressed by ESOP1, ESOP2and the product thereof. The third term is generated only when thefactor of the polarization dependency is present in the ring resonatorand in the lead part to the light receiver.

[0041] The differential resonance characteristic obtained bydifferentiating the formula (4) is required to detect the resonancepoint.

[0042] The differential resonance characteristic is controlled to bezero by the closed loop system. The resonance point of ESOP1 is usedhere to detect the rotation. The differential resonance characteristicis decomposed into three components corresponding to ESOP1, ESOP2 andthe interference component as shown in FIG. 19, and the operation pointof the resonance point of ESOP1, i.e., the position at which thedifferential resonance characteristic is zero is deviated from ξ=2qπ(q:integer) due to the presence of ESOP2 and the interference component.In order to calculate the operation point, the formula (4) isdifferentiated by the phase corresponding to each term to obtain ξ tosatisfy the formula (5).

D ₁(ξ)+D ₂(ξ−Δβ′)+D ₃(ξ, ξ−Δβ′)=0  (5)

[0043] D_(i) (i=1 to 3) expresses the differentiation by ξ of the i-thterm in the formula (4), and Δβ′ is the difference in the propagationconstant between two ESOPs. With N as an integer, the deviation fromξ=2qπ (q:integer) converted in the angular velocity is expressed asfollows: $\begin{matrix}{{\Delta\Omega} = {\frac{c_{0}\lambda}{4\pi \quad {Lr}}\left( {\zeta - {2N\quad \pi}} \right)}} & (6)\end{matrix}$

[0044] where c₀, λ and r are the velocity of light in vacuum, thewavelength of the light source, and the radius of the ring resonator,respectively.

[0045] Since the difference at each operation point of the CW light andthe CCW light is used for the output to detect the rotation, theoperations of the formulae (4) to (6) are implemented for the CW lightand the CCW light, respectively, and the difference is defined as theerror of the gyro output.

[0046] As described above, in the analysis of the 90° splice, nopolarizer is inserted in the lead part of the ring resonator.

[0047] The transfer matrix of the polarizer is expressed as follows:$\begin{matrix}{P_{1} = {\begin{pmatrix}1 & 0 \\0 & ɛ_{i}\end{pmatrix}\left( {{i = 1},2} \right)}} & (7)\end{matrix}$

[0048] The polarization dependency loss is similarly expressed.

[0049] Thus, ε₁=ε₂=1, and the third term in the formula (4) is notgenerated. Thus, the calculated error is affected only by ESOP2.

[0050] However, it is impossible that the polarization dependency lossis completely not present. The effect can be checked by substituting thenumerical value such as 0.9999 which is slightly changed from 1 into ε₁(i=1, 2).

[0051]FIG. 20 shows the errors of the gyro output calculated bysubstituting 1, 0.9999, 0.999 and 0.99 in ε₁.

[0052] It is understood from FIG. 20 that the errors of the gyro outputare rapidly increased by the slight change in ε₁. It is thus understoodthat the calculated error with ε_(i)=1 for a condition in which nopolarizer is used, i.e., the accuracy of the gyro cannot be expected inpractice.

[0053] Separate from the present invention, the countermeasures havebeen tried for the above problems, and disclosed in U.S. Pat. No.5,296,912, Strandjord et al., “R-FOG ROTATION RATE ERROR REDUCER HAVINGRESONATOR MODE SYMMETRIZATION” (Literature 7) and L. K. Strandjord andG. A. Sanders, “PERFORMANCE IMPROVEMENTS OF A POLARIZATION-ROTATINGRESONATOR FIBER OPTIC GYROSCOPE”, Proc. SPIE, Fiber Optic and LaserSensors X, Vol. 1795, pp. 94-104, 1992 (Literature 8).

[0054] In the countermeasures, the error can be reduced if the frequencyis switched to allow the frequency of oscillation of the light source toalternately follow ESOP1 and ESOP2, and the output generated in eachcase is averaged by using the different sign of the error when ESOP1 isused for detecting the rotation, and when ESOP2 is used for detectingthe rotation.

[0055] In addition, the countermeasures that the output of the ringresonator observed by the light receiver has no errors is disclosed inL. Strandjord and G. Sanders, “PASSIVE STABILIZATION OF TEMPERATUREDEPENDENT POLARIZATION ERRORS OF A POLARIZATION-ROTATING RESONATOR FIBEROPTIC GYROSCOPE”, Proc. SPIE, Fiber Optic and Laser Sensors XIII, Vol.2510, pp. 81-91, 1995 (Literature 9). The countermeasures were verifiedfor the transmitter ring resonator.

[0056] In this countermeasures, the error is reduced if the differencebetween the length L₁ from the first coupler to the 90° splice point andthe length L₂ from the second coupler and the 90° splice point is zero,and the length L₃ between the first coupler and the second coupler isclose to zero, where L₁, L₂ and L₃ are the lengths of three portions ofthe optical fiber divided by the first coupler in which the light waveemitted from the laser beam source first reaches, the 90° splice pointin the ring resonator, and the second coupler to emit the light wave andinput the light wave in the light receiver, respectively.

[0057] However, in the theoretical development leading to the result, alarge number of approximations are used, and a focus is placed in thespace between the resonance point of ESOP1 and the resonance point ofESOP2, i.e., in that one resonance point is located in the center of theother repeatedly appearing resonance point; however, the mechanism ofgeneration of the error appearing in the gyro output is not analyzed.

SUMMARY OF THE INVENTION

[0058] The present invention has been made to solve various problemsdescribed above, and an object of the present invention is therefore toprovide a method of eliminating the error generated in the output of aresonator fiber optic gyro even when the loss in the polarizationdependency which has never been solved in conventional methods ispresent in a ring resonator.

[0059] A method of reducing the polarization fluctuation inducing driftin a resonator fiber optic gyro includes the step of setting ΔL so thatthe relationship of ΔL and Δβ satisfies a formula ΔβΔL=π+2nπ [radian](n: integer), or approximately satisfies the formula to minimize theerror induced by the polarization fluctuation where ΔL is the differencein length between L₁ and L₂ of two portions of a waveguide divided by acoupler and the polarization-rotating point, and Δβ is the difference inpropagation constant of two axes of polarization of the waveguide in amethod of measuring the non-reciprocal effect such as the rotation in areflector ring resonator comprising a sensing loop formed of thewaveguide having two axes of polarization to propagate the light wave,and the coupler which is inserted in said sensing loop, guides the lightwave from a laser beam source to said sensing loop and emits the lightwave in said sensing loop, and having a polarization-rotating point insaid sensing loop.

[0060] The method of reducing the polarization fluctuation inducingdrift in a resonator fiber optic gyro includes the step of setting ΔLand L₃ so that the relationship of ΔL, Δβ and L₃ satisfies a formulaΔβΔL=π+2nπ [radian] (n: integer) and ΔβL₃=m π [radian] (m: integer), orapproximately satisfies the formulae to minimize the error induced bythe polarization fluctuation where L₁ is the distance from said firstcoupler to the polarization-rotating point, L₂ is the distance from thepolarization-rotating point to said second coupler, L₃ is the distancefrom said second coupler to said first coupler, ΔL is the differencebetween L₁ and the length (L₂+L₃) from the polarization-rotating pointto said first coupler through said second coupler, and Δβ is thedifference in propagation constant of two axes of polarization of thewaveguide when the waveguide is divided into three portions by saidfirst coupler, said polarization-rotating point and said second couplerin a method of measuring the non-reciprocal effect such as the rotationin a transmitter ring resonator comprising a sensing loop formed of thewaveguide having two axes of polarization to propagate the light wave, afirst coupler to guide the light wave from a laser beam source to saidsensing loop and a second coupler to emit the light wave in said sensingloop which are inserted in said sensing loop, and having apolarization-rotating point in said sensing loop.

[0061] In the present invention, the polarization-rotating angle at thepolarization-rotating point can be set to be about 90° in the aboveconfiguration.

[0062] The method of reducing the polarization fluctuation inducingdrift in a resonator fiber optic gyro includes the step of minimizingthe error irrespective of any change in ΔβL by controlling ΔL making useof the fact that the characteristic of said measurement error isconsiderably dependent on the product of Δβ and ΔL where ΔL is thedifference in length of two portions of a waveguide divided by saidfirst coupler and said polarization-rotating point, and Δβ is thedifference in propagation constant of two axes of polarization of thewaveguide, and less dependent on ΔβL which is the product of the sum ofthe length of two portions of the waveguide (L) and Δβ which is the sumin propagation constant of two axes of polarization of the waveguide ina method of measuring the non-reciprocal effect such as the rotation ina reflector ring resonator or a transmitter ring resonator comprising asensing loop formed of the waveguide having two axes of polarization topropagate the light wave and a coupler inserted in said sensing loop,and having a polarization-rotating point in said sensing loop.

[0063] In the present invention, the error is minimized irrespective ofany change in ΔβL through the feedback to the difference ΔL in lengthbetween two portions of a waveguide divided by a coupler in which thelight wave emitted from a laser beam source reaches first and thepolarization-rotating point in a ring resonator making use of generationof an error signal indicating the deviation from an optimum value ofΔβΔL at a predetermined period by alternately applying two differentdepths of modulation in the bias modulation implemented for detecting aresonance point at the predetermined period.

[0064] In the above configuration, the ring resonator itself can be setin a condition in which no errors are generated even when thepolarization dependency loss is present in the ring resonator. Thus, theresonator fiber optic gyro using this method can suppress the errors ofthe gyro output induced by the polarization fluctuation to a minimum,and reduce the drift of the gyro output caused by the change in theerrors. The gyro high in accuracy can thus be realized.

BRIEF DESCRIPTION OF THE DRAWINGS

[0065] In the accompanying drawings:

[0066]FIG. 1 is a schematic representation for explaining a firstembodiment of a resonator fiber optic gyro in accordance with thepresent invention, and an overall view of a reflector ring resonatormodel;

[0067]FIG. 2 is a schematic representation for explaining a calculatingmethod of a resonance point from the resonance characteristic;

[0068]FIG. 3A is a graph showing the result of the numerical simulationof the errors using a reflector ring resonator with ε₁=ε₂=0.01;

[0069]FIG. 3B is a graph showing the result of the numerical simulationof the errors using the reflector ring resonator with ε₁=ε₂=0.99;

[0070]FIG. 3C is a graph showing the mode of generation of the errors toΔβΔL;

[0071]FIG. 3D is a graph showing the result of the numerical simulationof the errors when the polarization dependency is assumed only in acoupler with κ_(x)≠κ_(y) γ_(x)≠γ_(y), and ε₁=ε₂=1;

[0072]FIG. 3E is a graph showing the result of the numerical simulationof the errors when the polarization dependency is assumed in a lead partand the coupler with κ_(x)≠κ_(y), γ_(x)≠γ_(y), and ε₁=ε₂=0.01;

[0073]FIG. 3F is a graph showing the result of the numerical simulationof the errors when the polarization dependency is assumed in the leadpart and the coupler with κ_(x)≠κ_(y), γ_(x)≠γ_(y), and ε₁=ε₂=0.99;

[0074]FIG. 3G is a graph showing the result of the analysis when thesplice angle θ in the ring resonator is set to be θ=90°;

[0075]FIG. 3H is a graph showing the result of the analysis when thesplice angle θ in the ring resonator is set to be θ=75°;

[0076]FIG. 31 is a graph showing the result of the analysis when thesplice angle θ in the ring resonator is set to be θ=45°;

[0077]FIG. 3J is a graph showing the result of the analysis when thesplice angle θ in the ring resonator is set to be θ=10°;

[0078]FIG. 4 is a graph showing a schematic representation forexplaining a second embodiment of the resonator fiber optic gyro inaccordance with the present invention, and an overall view of thetransmitter ring resonator model;

[0079]FIG. 5A is a graph showing the result of the numeral simulation ofthe errors when the transmitter ring resonator is used, and nopolarization dependency is present in the output side coupler and theoutput side lead part;

[0080]FIG. 5B is a graph showing the result of the numeral simulation ofthe errors when the transmitter ring resonator is used, and thepolarization dependency is present in the output side coupler and theoutput side lead part;

[0081]FIG. 5C is a graph showing the result of the numeral simulation ofthe errors when the transmitter ring resonator is used; i.e., the resultto ΔβL and Δβ(L₁-L₂) when the polarization dependency is present in theoutput side coupler and the output side lead part, and ΔβL₃=π;

[0082]FIG. 5D is a graph showing the result of the numeral simulation ofthe errors when the transmitter ring resonator is used; i.e., the resultto ΔβL and Δβ(L₁-L₂) when the polarization dependency is present in theoutput side coupler and the output side lead part, and ΔβL₃=2π;

[0083]FIG. 5E is a graph showing the result of the numerical simulationof the errors for two different depths of modulation, and shows thecomparison of the errors to ΔβΔL;

[0084]FIG. 5F is a graph showing the result of the numerical simulationof the errors for two different depths of modulation, and shows thecomparison of the errors to ΔβΔL taking into consideration the signthereof;

[0085]FIG. 6 is a graph showing the result of the numerical simulationof the errors in the reflector ring resonator model when the incidentpolarization is deviated from the straight polarization;

[0086]FIG. 7 shows a first embodiment of a resonator fiber optic gyro inaccordance with the present invention, which is the reflector ringresonator;

[0087]FIG. 8 shows a second embodiment of the resonator fiber optic gyroin accordance with the present invention, in which the position of theoutput side coupler of the transmitter ring resonator is optimized;

[0088]FIG. 9 shows the reflector and transmitter ring resonators;

[0089]FIG. 10 shows the resonance characteristic to the frequency of thelight wave of the ring resonator in FIG. 9;

[0090]FIG. 11 shows the resonance characteristic to the phase differenceof the light waves different in periodicity by one trip of the ringresonator in FIG. 9;

[0091]FIG. 12 is a block diagram showing the configuration of thereflector-resonator fiber optic gyro;

[0092]FIG. 13 is a block diagram showing the configuration of thetransmitter-resonator fiber optic gyro;

[0093]FIG. 14 shows the ring resonator using a polarization maintainingfiber and the splice position;

[0094]FIG. 15 is a conceptual view showing the Eigenstate ofPolarization (ESOP);

[0095]FIG. 16 is a schematic representation explaining the 90° splice inthe ring resonator using the polarization maintaining fiber;

[0096]FIG. 17 shows the resonance characteristic when the polarizationis rotated by 90° in the ring resonator;

[0097]FIG. 18 shows the ring resonator (reflector type) using thepolarization maintaining fiber;

[0098]FIG. 19 is a graph explaining ESOP1, ESOP2 and interferencecomponent in the resonance differential characteristic;

[0099]FIG. 20 shows the rapid increase of the errors due to the slightpolarization dependency;

[0100]FIG. 21 is a conceptual view of removing the errors using twodepths of modulation;

[0101]FIG. 22 is a block diagram (reflector type) of R-FOG applying thecontrol of ΔβΔL; and

[0102]FIG. 23 is a block diagram (transmitter type) of R-FOG applyingthe control of ΔβΔL.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0103] The preferred embodiments of the present invention will bedescribed in detail with reference to the drawings.

[0104] In order to find out a method of analyzing the errors generatedin measuring the non-reciprocal effect such as the rotation to preventthe errors in a ring resonator comprising a wave guide to propagate thelight wave and a coupler, it is necessary to consider the presence ofthe polarization dependency loss in a lead part from the ring resonatorto a light receiver.

[0105] In the present invention, a method of minimizing the errors isfound out by assuming the polarization dependency loss to each componentof the ring resonator to obtain the resonance characteristic of the CWlight and CCW light, calculating the deviation (the operation point) ofthe resonance point of each ESOP1, and analyzing the mechanism ofgeneration of the errors appearing in the gyro output by the differencetherebetween.

[0106] Most important parameters in the analysis of the presentinvention include ΔβL and ΔβΔL to indicate that the sensing loop lengthof the ring resonator is expandable by the change in the environmentaltemperature. Δβ is the differential propagation constant of twopolarization axes of the polarization maintaining fiber, and L and ΔLare the sum and the difference of the lengths L₁ and L₂ of two portionsof the wave guide to be split at the splice position in the coupler atwhich the light wave emitted from the laser beam source arrives firstand the ring resonator.

[0107] The present invention verifies how both ΔβL and ΔβΔL are involvedin generation of the errors in order to clarify the generation mechanismof the errors. Both ΔβΔL and ΔβL have the periodicity of 2π, and thetotal image of the error generation can be grasped if the errors arecalculated from 0 to 2π. Since the effect by ΔβL appears symmetric withrespect to π, ΔβL is set to be between 0 and π in the analysis of thepresent invention.

[0108] In every analysis for the present invention, the parameters areset to the following values: Sensing loop length=15 m, sensing loopdiameter r=0.05 m, wavelength λ of laser beam source=1.3 μm, spliceangle θ in the ring resonator =89°, angular deviations with respect tofiber polarization axes of the incident linearly polarized waveθ_(iCW)=1° and θ_(iCCW)=2°.

[0109] Other parameters used in the analysis of the present inventionare as follows:

[0110] Splice angle in ring resonator: θ [°]

[0111] Polarization dependency loss (reflector) at lead part: ε_(i)(i=1-2)

[0112] Polarization dependency loss (transmitter) at lead part: ε_(i)(i=1-4)

[0113] Coupling ratio at coupler (reflector): κ_(x), κ_(y)

[0114] Loss at coupler (reflector): γ_(x), γ_(y)

[0115] Crosstalk at coupler (reflector): θ_(R)[°]

[0116] Coupling ratio at input side coupler (transmitter): κ_(1x),κ_(1y)

[0117] Coupling ratio at output side coupler (transmitter): κ_(2x),K_(2y)

[0118] Loss at input side coupler (transmitter): γ_(1x), γ_(1y)

[0119] Loss at output side coupler (transmitter): γ_(2x)γ_(2y)

[0120] Crosstalk at input side coupler (transmitter): θ_(R1) [°]

[0121] Crosstalk at output side coupler (transmitter): θ_(R2) [°]

[0122] (Reflector ring resonator)

[0123] The analysis of the errors when using the reflector ringresonator for the resonator fiber optic gyro with the method inaccordance with the present invention applied thereto will be describedbelow.

[0124]FIG. 1 shows the model of the reflector ring resonator used in theanalysis of the present invention.

[0125] The coupler is assumed to have the polarization dependency, andthe reflection matrix C_(r) indicating the reflection by the coupler isexpressed as follows: $\begin{matrix}{C_{r} = \begin{pmatrix}\sqrt{1 - \kappa_{x}} & 0 \\0 & \sqrt{1 - \kappa_{y}}\end{pmatrix}} & (8)\end{matrix}$

[0126] The transmission matrix C_(t) indicating the transmission by thecoupler is expressed as follows: $\begin{matrix}{C_{t} = {\begin{pmatrix}{j\sqrt{\kappa_{x}}} & 0 \\0 & {j\sqrt{\kappa_{y}}}\end{pmatrix}\begin{pmatrix}{\cos \left( \theta_{R} \right)} & {- {\sin \left( \theta_{R} \right)}} \\{- {\sin \left( \theta_{R} \right)}} & {- {\sin \left( \theta_{R} \right)}}\end{pmatrix}}} & (9)\end{matrix}$

[0127] The matrix C_(l) indicating the loss is expressed as follows:$\begin{matrix}{C_{1} = \begin{pmatrix}\sqrt{1 - r_{x}} & 0 \\0 & \sqrt{1 - r_{y}}\end{pmatrix}} & (10)\end{matrix}$

[0128] The splice in the ring resonator is expressed as follows:$\begin{matrix}{{R(\theta)}\begin{pmatrix}{\cos (\theta)} & {- {\sin (\theta)}} \\{\sin (\theta)} & {\cos (\theta)}\end{pmatrix}} & (11)\end{matrix}$

[0129] where θ is the angle of rotation.

[0130] The transfer matrix of the fiber is given by the formula (12)using this rotation.

A(φ)=exp{−jξ(z/L)}exp(−iβ _(AV) z)C(φ)  (12)

[0131] where ${C(\phi)} = \begin{bmatrix}C_{11} & C_{12} \\C_{21} & C_{22}\end{bmatrix}$

[0132] C₁₁C₂₂*=cos (ηz)−j{Δβ/(2η)} sin (ηz)

[0133] C₁₂=−C₂₁=(φ/η) sin (ηz)

[0134] η={square root}{square root over ((Δβ/2)²+φ²)}

[0135] β_(AV)=(β_(x)+β_(y))/2

[0136] Δβ=β_(x)−β_(y)

[0137] φ=θ/L

[0138] where β_(i) (i=x, y) is the propagation constant of twopolarization axes of the polarization maintaining fiber, and Δβis thedifference in propagation constant of two polarization axes of thepolarization maintaining fiber. ξ is the parameter used in the analysis,and indicates the phase change, i.e., that the phase difference isgenerated between two light waves of the polarization maintaining fiberpropagating in the ring resonator and different in periodicity by onetrip.

[0139] Using the formulae (11) and (12), the transfer matrix F_(bCW) ofthe sensing loop of the CW light is expressed as follows:$\begin{matrix}\begin{matrix}{F_{bCW} = {{A(\phi)}{_{Z = L_{2}}{\cdot {R(\theta)} \cdot {A(\phi)}}}_{Z = L_{1}}}} \\{= {{\exp \left( {{- j}\quad \xi} \right)}{{\exp \left( {{- j}\quad \beta_{A\quad V}L} \right)}\begin{bmatrix}t_{11{CW}} & t_{12{CW}} \\t_{21{CW}} & t_{22{CW}}\end{bmatrix}}}}\end{matrix} & (13)\end{matrix}$

[0140] Using this, the transfer matrix Tcw of the CW light making onetrip around the ring resonator of is expressed as follows:

T_(CW)C₁C_(r)F_(bCW)  (14)

[0141] where the loss in the sensing loop is included in the loss in thecoupler. The eigenvalues of T_(CW) are kicw and λ_(2CW), and theeigenvector to kXcw is given by: $\begin{matrix}\begin{bmatrix}X_{1{CW}} \\Y_{1{CW}}\end{bmatrix} & (15)\end{matrix}$

[0142] and the eigenvector to λ_(2CW) is given by: $\begin{matrix}\begin{bmatrix}X_{2{CW}} \\Y_{2{CW}}\end{bmatrix} & (16)\end{matrix}$

[0143] Using the eigenvalues and eigenvectors, T_(CW) is rewritten asfollows:

T _(CW) =Z _(CW) Λ _(CW) Z _(CW) ⁻¹  (17)

[0144] where $\begin{matrix}{Z_{CW} = \begin{pmatrix}X_{1{CW}} & X_{2{CW}} \\Y_{1{CW}} & Y_{2{CW}}\end{pmatrix}} & (18) \\{\Lambda_{CW} = \begin{pmatrix}\lambda_{1{CW}} & 0 \\0 & \lambda_{2{CW}}\end{pmatrix}} & (19)\end{matrix}$

[0145] From the following relationship,

T _(CW) ^(m)=(Z _(CW)Λ_(CW) Z _(CW) ⁻¹)^(m) =Z _(CW)Λ_(CW) ^(m) Z _(CW)⁻¹  (20)

[0146] $\begin{matrix}\begin{matrix}{{\sum\limits_{n = 0}^{\infty}T_{CW}^{n + 1}} = {\sum\limits_{n = 0}^{\infty}\left( {Z_{CW}\Lambda_{CW}^{n + 1}Z_{CW}^{- 1}} \right)}} \\{= {{Z_{CW}{\sum\limits_{n = 0}^{\infty}{\Lambda_{CW}^{n + 1}Z_{CW}^{- 1}}}} = {Z_{CW}{\sum\limits_{n = 0}^{\infty}{\begin{pmatrix}\lambda_{1{CW}}^{n + 1} & 0 \\0 & \lambda_{2{CW}}^{n + 1}\end{pmatrix}Z_{CW}^{- 1}}}}}} \\{= {{{Z_{CW}\begin{pmatrix}{\sum\limits_{n = 0}^{\infty}\lambda_{1{CW}}^{n + 1}} & 0 \\0 & {\sum\limits_{n = 0}^{\infty}\lambda_{2{CW}}^{n + 1}}\end{pmatrix}}Z_{CW}^{- 1}} = {{Z_{CW}\begin{pmatrix}\frac{\lambda_{1{CW}}}{1 - \lambda_{1{CW}}} & 0 \\0 & \frac{\lambda_{2{CW}}}{1 - \lambda_{2{CW}}}\end{pmatrix}}Z_{CW}^{- 1}}}} \\{= {Z_{CW}\Gamma_{CW}Z_{CW}^{- 1}}}\end{matrix} & (21)\end{matrix}$

[0147] where $\begin{matrix}{\Gamma_{CW} = \begin{pmatrix}\frac{\lambda_{1{CW}}}{1 - \lambda_{1{CW}}} & 0 \\0 & \frac{\lambda_{2{CW}}}{1 - \lambda_{2{CW}}}\end{pmatrix}} & (22)\end{matrix}$

[0148] The transfer matrix T_(CCW) of the CCW light making one triparound the ring resonator of is expressed as follows:

T_(CCW)=C_(l)C_(r)F_(bCCW F) _(bCCW)=F_(bCW) ^(†)  (23)

[0149] where † indicates the adjoint matrix.

[0150] The eigenvalues λ_(1CCW) and λ_(2CCW) of T_(CCW) are related toeach other by the following expression: $\begin{matrix}{\begin{bmatrix}\lambda_{1{CCW}} \\\lambda_{2{CCW}}\end{bmatrix} = \begin{bmatrix}\lambda_{1{CW}} \\\lambda_{2{CW}}\end{bmatrix}} & (24)\end{matrix}$

[0151] The eigenvector for λ_(1CCW) is expressed as follows:$\begin{matrix}{\begin{bmatrix}X_{1{CCW}} \\Y_{1{CCW}}\end{bmatrix} = \begin{bmatrix}X_{1{CW}}^{*} \\Y_{1{CW}}^{*}\end{bmatrix}} & (25)\end{matrix}$

[0152] The eigenvector for λ_(2CCW) is expressed as follows:$\begin{matrix}{\begin{bmatrix}X_{2{CCW}} \\Y_{2{CCW}}\end{bmatrix} = \begin{bmatrix}X_{2{CW}}^{*} \\Y_{2{CW}}^{*}\end{bmatrix}} & (26)\end{matrix}$

[0153] Then, T_(CCW) is rewritten as follows:

T _(CCW) =Z _(CCW)Λ_(CCW) Z _(CCW) ⁻¹  (27)

[0154] where $\begin{matrix}{Z_{CCW} = \begin{pmatrix}X_{1\quad {CCW}} & X_{2\quad {CCW}} \\Y_{1\quad {CCW}} & Y_{2\quad {CCW}}\end{pmatrix}} & (28) \\{\Lambda_{CCW} = \begin{pmatrix}\lambda_{1\quad {CCW}} & 0 \\0 & \lambda_{2\quad {CCW}}\end{pmatrix}} & (29)\end{matrix}$

[0155] Similar to CW, the following relationship is derived.$\begin{matrix}{{\sum\limits_{n = 0}^{\infty}T_{CCW}^{n + 1}} = {Z_{CCW}\Gamma_{CCW}Z_{CCW}^{- 1}}} & (30)\end{matrix}$

[0156] where $\begin{matrix}{\Gamma_{CCW} = \begin{pmatrix}\frac{\lambda_{1{CCW}}}{1 - \lambda_{1\quad {CCW}}} & 0 \\0 & \frac{\lambda_{2\quad {CCW}}}{1 - \lambda_{2{CCW}}}\end{pmatrix}} & (31)\end{matrix}$

[0157] The light wave emitted from the laser beam source is the linearlypolarized wave, and assumed to be incident on X-axis of the polarizationmaintaining fiber.

[0158] The incident light E_(0CW) is expressed as follows:$\begin{matrix}{E_{0{CW}} = \begin{pmatrix}{\cos \left( \theta_{iCW} \right)} \\{\sin \left( \theta_{iCW} \right)}\end{pmatrix}} & (32)\end{matrix}$

[0159] where θ_(iCW) is the angular deviation to X-axis of thepolarization maintaining fiber.

[0160] E_(0CW) firstly passes through a polarizer P₁ of the lead part.P₁ is expressed as follows: $\begin{matrix}{P_{1} = \begin{pmatrix}1 & 0 \\0 & ɛ_{1}\end{pmatrix}} & (33)\end{matrix}$

[0161] The light wave Eicw passing through the polarizer P₁ is expressedas follows:

E_(1CW)P₁E_(oCW)  (34)

[0162] Using E_(1CW) and the formula (21), the light wave E_(2CW)inputted in the coupler with the CW light making trips around the ringresonator is expressed as follows: $\begin{matrix}\begin{matrix}E_{2{CW}} & {= {F_{bCW}{\sum\limits_{n = 0}^{\infty}{T_{CW}^{n}C_{1}C_{t}E_{1{CW}}}}}} \\\quad & {= {\left( {C_{1}C_{r}} \right)^{- 1}T_{CW}{\sum\limits_{n = 0}^{\infty}{T_{CW}^{n}C_{1}C_{t}E_{1{CW}}}}}} \\\quad & {= {\left( {C_{1}C_{r}} \right)^{- 1}{\sum\limits_{n = 0}^{\infty}{T_{CW}^{n + 1}C_{1}C_{t}E_{1{CW}}}}}} \\\quad & {= {\left( {C_{1}C_{r}} \right)^{- 1}Z_{CW}\Gamma_{CW}Z_{CW}^{- 1}C_{1}C_{t}E_{1{CW}}}}\end{matrix} & (35)\end{matrix}$

[0163] Using E_(1CW) and E_(2CW), the light wave E_(3CW) outputted fromthe ring resonator is expressed as follows:

E _(3CW) C _(l)(C _(r) E _(1CW) +C _(t) E _(2CW))  (36)

[0164] This light wave passes through a polarizer P₂ of the lead part.P₂ is expressed as follows: $\begin{matrix}{P_{2} = \begin{pmatrix}1 & 0 \\0 & ɛ_{2}\end{pmatrix}} & (37)\end{matrix}$

[0165] The light wave E_(dCW) inputted in a light receiver is expressedas follows:

E_(dDCW)=P₂E_(3CW)  (38)

[0166] The light reception intensity of the CW light observed by thelight receiver is expressed as follows:

|E_(dCW)|²=E_(dCW) ^(†)E_(dCW)  (39)

[0167] Similar to CW, the incident light E_(0CCW) is expressed asfollows: $\begin{matrix}{E_{0{CCW}} = \begin{pmatrix}{\cos \left( \theta_{iCCW} \right)} \\{\sin \left( \theta_{iCCW} \right)}\end{pmatrix}} & (40)\end{matrix}$

[0168] where θ_(iCCW) is the angular deviation to X-axis of thepolarization maintaining fiber. The incident light passes through thepolarizer P₂, and the light wave after passing through the polarizer isexpressed as follows:

E_(3CCW)=P₂E_(0CCW)  (41)

[0169] Using E_(3CCW) and the formula (30), the light wave E_(4CCW)inputted in the coupler with the CW light making trips around the ringresonator is expressed as follows: $\begin{matrix}\begin{matrix}{E_{4{CCW}} = {F_{bCCW}{\sum\limits_{n = 0}^{\infty}{T_{CCW}^{n}C_{1}C_{t}E_{3{CCW}}}}}} \\{= {\left( {C_{1}C_{r}} \right)^{- 1}Z_{CCW}\Gamma_{CCW}Z_{CCW}^{- 1}C_{1}C_{t}E_{3{CCW}}}}\end{matrix} & (42)\end{matrix}$

[0170] Using E_(3CCW) and E_(4CCW), the light wave E_(1CCW) outputtedfrom the ring resonator is expressed as follows:

E _(1CCW) =C _(l)(C _(r) W _(3CCW) +C _(t) E _(4CCW))  (43)

[0171] This light wave passes through the polarizer P₁ of the lead part,and the light wave E_(dCCW) inputted in the light receiver is expressedas follows:

E_(dCCW)=P₁E_(1CCW)  (44)

[0172] The light reception intensity of the CCW light observed by thelight receiver is expressed as follows:

|E_(dCCW)|²=E_(dCCW) ^(†)C_(dCCW)  (45)

[0173] The resonance characteristics of the CW light and CCW light areobtained from the formulae (39) and (45) by changing ξ in the formulae(12) and (13). This means that the light reception intensity of the CWlight and CCW light are respectively expressed as the function of ξ asfollows:

|E_(dCW)|²=I_(dCW)(ξ)

|E_(dCW)|²=I_(dCCW)(ξ)  (46)

[0174] FIGS. 2(a) and 2(b) are conceptual views showing the calculationmethods of the resonance characteristics and the resonance points of theCW light and CCW light with respect to ξ, respectively.

[0175] The resonance points used in this analysis are calculated bychanging ξ around the resonance point as shown FIG. 2(b) taking intoconsideration the bias modulation around the resonance point, anddetecting u at which the light reception intensity around the ξ value isequal to each other. This means that σ_(cw) to satisfy the followingformula is calculated as the position of the resonance point of CW.

I _(dCW)(σ_(CW) −Δξ _(CW))=I _(dCW)(σ_(CW) −Δξ _(CW))  (47)

[0176] Similarly, σ_(CCW) to satisfy the following formula is calculatedas the position of the resonance point of CCW.

I _(dCCW)(σ_(CCW) −Δξ _(CCW))=I _(dCCW)(σ _(CCW) +Δξ _(CCW))  (48)

[0177] Δξ_(CW) and Δξ_(CCW) express the depth of modulation of the biasmodulation to the CW light and CCW light, respectively, and are set tobe Δξ_(CW)=Δξ_(CCW), and half width at half maximum at the resonancepoint.

[0178] The error corresponds to the difference between σ_(CW) andσ_(CCW), and the error of the gyro output is given as follows similar tothe formula (6): $\begin{matrix}{{\Delta\Omega} = {\frac{c_{0}\lambda}{4\pi \quad L\quad r}\left( {\sigma_{CW} - \sigma_{CCW}} \right)}} & (49)\end{matrix}$

[0179]FIGS. 3A to 3J show the results of the numerical simulation of theerrors when using the reflector ring resonator.

[0180]FIG. 3A shows the result of the numerical simulation of the errorwhen ε₁=ε₂=0.01 assuming the polarizer at the lead part. Otherparameters include θ=89°, κ_(x)=κ_(y)=0.02, γ_(x)=γ_(y)=0.02, andθ_(R)=0°.

[0181] In the figure, it is shown that the error is minimized from thefollowing relationship.

ΔβΔL=π+2nπ[radian](n: integer)  (A)

[0182]FIG. 3B shows the result of the numerical simulation of the errorwhen ε₁=ε₂=0.99 assuming the loss of polarization dependency at the leadpart. Other parameters are same as those in FIG. 3A. In the figure, itis shown that the error is minimized from the relationship of theformula (A) similar to FIG. 3A.

[0183] It is shown from these findings that the mode of generation ofthe error to ΔβΔL as shown in FIG. 3C taking into consideration the signof the error means the oscillation with the period of 2π.

[0184]FIG. 3D shows the error when the polarization dependency ispresent only in the coupler with ε₁=ε₂=1 where no loss is present in thepolarizer or polarization dependency at the lead part. This shows theresult of the numerical simulation of the error with κ_(x)≠κ_(y),γ_(x)≠γ_(y). Other parameters include θ=89°, κ_(x)=0.02, κ_(y)=0.022,γ_(x)=0.02, γ_(y)=0.022, and θ_(R)=20. It is shown from the figures thatthe error is minimized from the relationship in the formula (A) similarto the cases of FIGS. 3A and 3B.

[0185] From the results in FIGS. 3A, 3B and 3D, it can be confirmed thatgeneration of the errors caused by the loss of the polarizationdependency has a same characteristic.

[0186]FIGS. 3E and 3F show the numerical simulation of the errors whenthe polarization dependency is assumed at the lead part and the coupler.The polarization dependency at the lead part is set to be ε₁ε₂=0.01 andε₁ε₂=0.99, respectively. Other parameters include θ=89°, κ_(x)=0.02,κ_(y)=0.022, γ_(x)=0.02, γ_(y)=0.022, and θ_(R)=2° for each case. It isshown from the figures that the error is minimized from the relationshipin the formula (A) similar to the cases of FIGS. 3A, 3B and 3D.

[0187] It is understood from the above results that the error generatedby the polarization dependency loss is minimized when satisfying therelationship of the formula (A). In addition, the error is substantiallyconstant to ΔβL for the predetermined ΔβΔL. This means that the error ofthe gyro output is very small in dependency of ΔβL, and dependent onΔβΔL.

[0188] ΔβL is changed by at least π in about 1° C., and considerablyaffected by the environmental temperature, and the effect of theenvironmental temperature on ΔβΔL can be kept very small by reducing ΔLwhile satisfying the relationship of the formula (A).

[0189] Thus, the error of the gyro output can be reduced irrespective ofthe environmental temperature by using the relationship of the formula(A).

[0190]FIGS. 3G to 3J show the result of the analysis when the spliceangle θ in the ring resonator is changed such as θ=90°, 75°, 45° and10°. Other parameters include ε₁ε₂=0.01, ε_(x)=0.02, ε_(y)=0.022,γ_(x)=0.02, γ_(y)=0.022, and θ_(R)=2° for each case.

[0191] It is shown from the figures that the error is minimized whenΔβΔL satisfies the relationship of the formula (A).

[0192] (Transmitter Ring Resonator)

[0193] The analysis of the errors will be described below when using atransmitter ring resonator in the resonator fiber optic gyro with themethod in accordance with the present invention applied thereto.

[0194]FIG. 4 shows a model of the transmitter ring resonator used in theanalysis.

[0195] ΔL is defined here to be ΔL=L₁−(L₂+L₃). This is equal in that thesum and the difference of two portions of the optical fiber divided bythe model of the reflector ring resonator, the incident side coupler andthe 90° splice point are L and ΔL, respectively.

[0196] In FIG. 5, P₁ and P₂ of the input side lead part, and P₃ and P₄of the output side lead part are assumed to be the losses of thepolarization dependency.

[0197] The reflection matrix C_(1r) indicating the reflection of theinput side coupler is expressed as follows: $\begin{matrix}{C_{1r} = \begin{pmatrix}\sqrt{1 - \kappa_{1x}} & 0 \\0 & \sqrt{1 - \kappa_{1y}}\end{pmatrix}} & (50)\end{matrix}$

[0198] The transmission matrix C₁₁ indicating the transmission isexpressed by the formula (51). $\begin{matrix}{C_{1t} = {\begin{pmatrix}{j\sqrt{\kappa_{1x}}} & 0 \\0 & {j\sqrt{\kappa_{1x}}}\end{pmatrix}\begin{pmatrix}{\cos \left( \theta_{R!} \right)} & {- {\sin \left( \theta_{R1} \right)}} \\{- {\sin \left( \theta_{R1} \right)}} & {- {\sin \left( \theta_{R1} \right)}}\end{pmatrix}}} & (51)\end{matrix}$

[0199] The matrix C₁₁ indicating the loss is expressed as follows:$\begin{matrix}{C_{1l} = \begin{pmatrix}\sqrt{1 - r_{1x}} & 0 \\0 & \sqrt{1 - r_{1y}}\end{pmatrix}} & (52)\end{matrix}$

[0200] The reflection matrix C_(2r) indicating the reflection of theoutput side coupler is expressed as follows: $\begin{matrix}{C_{2r} = \begin{pmatrix}\sqrt{1 - \kappa_{2x}} & 0 \\0 & \sqrt{1 - \kappa_{2y}}\end{pmatrix}} & (53)\end{matrix}$

[0201] The transmission matrix C_(2t) indicating the transmission isexpressed by the formula (54). $\begin{matrix}{C_{2t} = {\begin{pmatrix}{j\sqrt{\kappa_{2x}}} & 0 \\0 & {j\sqrt{\kappa_{2x}}}\end{pmatrix}\begin{pmatrix}{\cos \left( \theta_{R2} \right)} & {- {\sin \left( \theta_{R2} \right)}} \\{- {\sin \left( \theta_{R2} \right)}} & {- {\sin \left( \theta_{R2} \right)}}\end{pmatrix}}} & (54)\end{matrix}$

[0202] The matrix C_(2l) indicating the loss is expressed as follows:$\begin{matrix}{C_{2l} = \begin{pmatrix}\sqrt{1 - r_{2x}} & 0 \\0 & \sqrt{1 - r_{2y}}\end{pmatrix}} & (55)\end{matrix}$

[0203] The rotation in the ring resonator is expressed similar to theformula (11) using the splice angle θ. The transfer matrix of the fiberis expressed similar to the formula (12).

[0204] The transfer matrix of the sensing loop of the CW light isexpressed as follows: $\begin{matrix}\begin{matrix}{{F_{bcw} = {{A(\phi)}{{_{z = L_{3}}{{\cdot C_{2l}}C_{2r}{A(\phi)}}}_{z = L_{2}} \cdot {R(\theta)} \cdot {A(\phi)}}}}}_{z = L_{1}} \\{= {\exp {\left\{ {{- {j\left( {\beta_{AV} + \xi} \right)}}L} \right\} \begin{bmatrix}t_{11{CW}} & t_{12{CW}} \\t_{21{CW}} & t_{22{CW}}\end{bmatrix}}}}\end{matrix} & (56)\end{matrix}$

[0205] Using this, the transfer matrix T_(CW) per trip of the ringresonator of the CW light is expressed as follows:

T_(CW)=C₁₁C_(1r)F_(bCW)  (57)

[0206] The transfer matrix T_(CCW) per trip of the ring resonator of theCCW light is expressed as follows:

T_(CCW)=C₁₁C_(1r)F_(bCCW)  (58)

F_(bCCW)=F_(bCW) ^(†)

[0207] The eigenvalue and the eigenvector are expressed similar to theformulae (21) and (30) as follows: $\begin{matrix}{{\sum\limits_{n = 0}^{\infty}T_{CW}^{n + 1}} = {z_{CW}\Gamma_{CW}z_{CW}^{- 1}}} & (59) \\{{\sum\limits_{n = 0}^{\infty}T_{CCW}^{n + 1}} = {z_{CCW}\Gamma_{CCW}z_{CCW}^{- 1}}} & (60)\end{matrix}$

[0208] where $\begin{matrix}{Z_{{CW}{({CCW})}} = \begin{pmatrix}X_{1{{CW}{({CCW})}}} & X_{2{{CW}{({CCW})}}} \\Y_{1{{CW}{({CCW})}}} & Y_{2{{CW}{({CCW})}}}\end{pmatrix}} & (61)\end{matrix}$

$\begin{matrix}{\Gamma_{{CW}{({CCW})}} = \begin{pmatrix}\frac{\lambda_{1{{CW}{({CCW})}}}}{1 - \lambda_{1{{CW}{({CCW})}}}} & 0 \\0 & \frac{\lambda_{2{{CW}{({CCW})}}}}{1 - \lambda_{2{{CW}{({CCW})}}}}\end{pmatrix}} & (62)\end{matrix}$

[0209] The incident light E_(0CW) of CW is expressed as follows:$\begin{matrix}{E_{0{CW}} = \begin{pmatrix}{\cos \left( \theta_{icw} \right)} \\{\sin \left( \theta_{icw} \right)}\end{pmatrix}} & (63)\end{matrix}$

[0210] where θ_(iCW) is the angular deviation to X-axis of thepolarization maintaining Wz4 fiber.

[0211] E_(0CW) firstly passes through the polarization dependency loss(polarizer) P₁ of the input side lead part. P₁ is expressed as follows:$\begin{matrix}{P_{1} = \begin{pmatrix}1 & 0 \\0 & ɛ_{1}\end{pmatrix}} & (64)\end{matrix}$

[0212] The present invention E_(1CW) passing through the polarizer P₁ isexpressed as follows:

E_(1CW)=P₁E_(0CW)  (65)

[0213] Using E_(1CW) and the formula (59), the light wave E_(2CW) to beinputted in the incident side coupler with the CW light making tripsaround the ring resonator is expressed as follows: $\begin{matrix}\begin{matrix}{E_{2{CW}} = {F_{bCW}{\sum\limits_{n = 0}^{\infty}\quad {T_{CW}^{n}C_{1l}C_{1t}C_{1{CW}}}}}} \\{= {\left( {C_{1l}C_{1r}} \right)^{- 1}T_{CW}{\sum\limits_{n = 0}^{\infty}\quad {T_{CW}^{n}C_{1l}C_{1t}E_{1{CW}}}}}} \\{= {\left( {C_{1l}C_{1r}} \right)^{- 1}{\sum\limits_{n = 0}^{\infty}\quad {T_{CW}^{n + 1}C_{1l}C_{1t}E_{1{CW}}}}}} \\{= {\left( {C_{1l}C_{1r}} \right)^{- 1}Z_{CW}\quad \Gamma_{CW}Z_{CW}^{- 1}C_{1l}C_{1t}E_{1{CW}}}}\end{matrix} & (66)\end{matrix}$

[0214] Using E_(1CW) and E_(2CW), the light wave E_(4CW) to be inputtedin the incident side coupler with the CW light making trips around thering resonator is expressed as follows:

E _(4CW) =C ₁₁(C ₁₁ E _(1CW) +C _(1r) E _(2CW))  (67)

[0215] In addition, using E_(4CW), the light wave E_(5CW) outputted fromthe ring resonator is expressed as follows:

E _(SCW) =A(φ)|_(Z=L) ₂ ·R(θ)·A(φ)|_(Z=L) ₁ E _(4CW)  (68)

[0216] This light wave passes through the polarization dependency loss(polarizer) P₃ at the output side lead part. P₃ is expressed as follows:$\begin{matrix}{P_{3} = \begin{pmatrix}1 & 0 \\0 & ɛ_{3}\end{pmatrix}} & (69)\end{matrix}$

[0217] The light wave E_(dCW) inputted in the light receiver isexpressed as follows:

E_(dCW)=P₃E_(5CW) (70)

[0218] The light reception intensity of the CW light observed by thelight receiver is expressed as follows:

|E_(dCW)|²=E_(dCW) ^(†)E_(dCW)  (71)

[0219] Similar to CW, the incident light E_(0CCW) is expressed asfollows: $\begin{matrix}{E_{0{CCW}} = \begin{pmatrix}{\cos \left( \theta_{iCCW} \right)} \\{\sin \left( \theta_{iCCW} \right)}\end{pmatrix}} & (72)\end{matrix}$

[0220] where θ_(iCCW) is the angular deviation to X-axis of thepolarization maintaining fiber.

[0221] E_(0CCW) firstly passes through the polarization dependency loss(polarizer) P₂ at the input side lead part. P₂ is expressed as follows:$\begin{matrix}{P_{2} = \begin{pmatrix}1 & 0 \\0 & ɛ_{2}\end{pmatrix}} & (73)\end{matrix}$

[0222] The light wave E3CCW passing through the polarizer P₂ isexpressed as follows:

E_(3CCW)=P₂E_(0CCW)  (74)

[0223] Using E_(3CCW) and the formula (60), the light wave E_(4CCW)inputted in the coupler with the CCW light making trips around the ringresonator is expressed as follows: $\begin{matrix}\begin{matrix}{E_{4{CCW}} = {F_{bCCW}{\sum\limits_{n = 0}^{\infty}\quad {T_{CCW}^{n}C_{1l}C_{1t}E_{3{CCW}}}}}} \\{= {\left( {C_{1l}C_{1r}} \right)^{- 1}T_{CCW}{\sum\limits_{n = 0}^{\infty}\quad {T_{CCW}^{n}C_{1l}C_{1t}E_{3{CCW}}}}}} \\{= {\left( {C_{1l}C_{1r}} \right)^{- 1}{\sum\limits_{n = 0}^{\infty}\quad {T_{CCW}^{n + 1}C_{1l}C_{1t}E_{3{CCW}}}}}} \\{= {\left( {C_{1l}C_{1r}} \right)^{- 1}Z_{CCW}\quad \Gamma_{CCW}Z_{CCW}^{- 1}C_{1l}C_{1t}E_{3{CCW}}}}\end{matrix} & (75)\end{matrix}$

[0224] Using E_(3CCW) and E_(4CCW), the light wave E_(2CCW) outputtedfrom the incident side coupler with the CCW light making trips aroundthe ring resonator is expressed as follows:

E _(2CCW) =C ₁₁(C ₁₁ E _(3CCW) +C _(1r) E _(4CCW))  (76)

[0225] In addition, using E_(2CCW), the light wave E_(5CCW) outputtedfrom the ring resonator is expressed as follows:

E_(5CCW)=C_(2l)C_(2t)A(φ)|_(Z=L) ₁ E_(2CCW)  (77)

[0226] This light wave passes through the polarization dependency lossP₄ at the output side lead part. P₄ is expressed as follows:$\begin{matrix}{P_{4} = \begin{pmatrix}1 & 0 \\0 & ɛ_{4}\end{pmatrix}} & (78)\end{matrix}$

[0227] The light wave E_(dCCW) inputted in the light receiver isexpressed as follows:

E_(dCCW)=P₄E_(5CCW)  (79)

[0228] The light reception intensity of the CCW light observed by thelight receiver is expressed as follows:

|E _(dCCW)|² =E _(dCCW) ⁺ E _(dCCW)  (80)

[0229] In the transmitter ring resonator, similar to the reflector ringresonator, the resonance characteristics of the CW light and the CCWlight can be obtained by the formulae (71) and (80) by changing in theformulae (12) and (56).

[0230] Similar to the reflector ring resonator, the resonance pointsσ_(CW) and σ_(CCW) of the CW light and the CCW light are obtainedrespectively from the formulae (47) and (48), and the errors arecalculated from the formula (49).

[0231]FIGS. 5A to 5F show the results of the numerical simulation of theerrors when using the transmitter ring resonator.

[0232]FIG. 5A shows the result of the numerical simulation of the errorwhen no polarization dependency is present at the output side couplerand the output side lead part, i.e., when κ_(2x)=κ_(2y)=0.02,γ_(2x)=γ_(2y)=0.02, θ_(R2)=0°, and ε₃=ε₄=1. Other parameters includeθ=89°, κ_(1x)=0.02, κ_(1y)=0.022 , γ_(1x)=0.02, γ_(1y)=0.02, andθ_(R1)=2°, and ε₁=ε₂=0.99.

[0233] It is understood from the figure that, similar to the reflectorring resonator, the error is minimized by the relationship of theformula (A).

[0234] It is thus proven that the relationship of the formula (A) isgenerally established irrespective of the reflector ring resonator orthe transmitter ring resonator.

[0235]FIG. 5B shows the analysis when the polarization dependency ispresent at both the output side coupler and the output side lead part.

[0236] The parameters include θ=89°, ε₁ε₂=0.99, κ_(1x)=0.02,κ_(1y)=0.022, γ_(1x)=0.02, γ_(1y)=0.022, θ_(R1)=2°, ε₃ε₄=0.99,κ_(2x)=0.02, κ_(2y)=0.022, γ_(2x)=0.02, γ_(2y)=0.022, and θ_(R2)=2°.

[0237] Deviation is present from the relationship of the formula (A).This means that generality of the formula (A) is maintained when nopolarization dependency is present or the polarization dependency issmall at the output side coupler or the output side lead part.

[0238] In order to remove the effect of the polarization dependency atthe output side coupler and the output side lead part, it is necessaryto regulate the optimum position of the output side coupler whilemaintaining the relationship of the formula (A). It will be describedbelow that the relationship of the formula (A) is applicable to thetransmitter ring resonator if the length of the input side coupler andthe output side coupler is integral multiple of π.

[0239]FIG. 5C, similar to FIG. 5B, shows the analysis in which thepolarization dependency is present at both the output side coupler andthe output side lead part. The same parameters as those in FIG. 5B areincluded. However, the analysis was implemented for ΔβL and Δβ(L₁-L₂)with ΔβL₃=π. It is understood from the figure that the error isminimized with Δβ(L₁-L₂)=2nπ (n: integer), and this relationship agreeswith the relationship of the formula (A) with respect to the sum and thedifference of two portions of the optical fiber divided by the incidentside coupler and the 90° splice point.

[0240]FIG. 5D, similar to FIGS. 5B and 5C, shows the analysis in whichthe polarization dependency is present at both the output side couplerand the output side lead part. The same parameters as those in FIGS. 5Band 5C are included. However, the analysis was implemented for ΔβL andΔβ(L₁-L₂) with ΔβL₃=2π.

[0241] It is understood from the figure that the error is minimized withαβ(L₁-L₂)=π+2nπ (n: integer), and this relationship agrees with therelationship of the formula (A) with respect to the sum and thedifference of two portions of the optical fiber divided by the incidentside coupler and the 90° splice point.

[0242] As described above, the error induced by the polarizationfluctuation if the following formulae are satisfied:

ΔβΔL=π+2nπ[radian] (n: integer)

[0243] and

ΔβL ₃ =mπ[radian] (m: integer)  (B)

[0244] where ΔL=L₁−(L₂+L₃)

[0245] The error induced by the polarization fluctuation can be reducedby setting AL and L₃ so as to satisfy the relationship. This is a methodof reliably reducing the error while maintaining the relationship of theformula (A) when using the transmitter ring resonator.

[0246] (Length of Lead Part)

[0247] In the above analysis, the light wave incident in the ringresonator is assumed to be the linearly polarized wave. However, if thelight wave incident in the ring resonator is not linearly polarizedwave, the result of the analysis is different. The error in thiscondition can be obtained by replacing the formulae (32) and (63) by theformula (82), and the formulae (63) and (72) by the formula (83),respectively. $\begin{matrix}{E_{0{CW}} = \begin{pmatrix}{\cos \left( \theta_{iCW} \right)} \\{{\sin \left( \theta_{iCW} \right)}{\exp \left( {i\quad \varphi_{1{CW}}} \right)}}\end{pmatrix}} & (82) \\{E_{0{CCW}} = \begin{pmatrix}{\cos \left( \theta_{iCCW} \right)} \\{{\sin \left( \theta_{1{CCW}} \right)}{\exp \left( {i\quad \varphi_{iCCW}} \right)}}\end{pmatrix}} & (83)\end{matrix}$

[0248] where φ_(iCW) and φ_(iCCW) denote the phase difference betweenthe light wave propagating along X-axis and the light wave propagatingalong Y-axis.

[0249]FIG. 6 shows the result of the numerical simulation of the errorwhen the incident polarized wave is deviated from the linearly polarizedwave in the reflector ring resonator model. Here, φ_(iCW)=0 andφ_(iCCW)=0.5π. Other parameters include θ=90°, ε₁=ε₂=0.01, κ_(x)=0.02,κ_(y)=0.022, γ_(x)=0.02, γ_(y)=0.022, and θ_(R)=0°.

[0250] It is understood from the figure that the value of ΔβΔL with theerror minimized thereat is deviated from the relationship of the formula(A) due to the deviation from the linearly polarized wave, and it isthus necessary that the light wave incident in the ring resonator isconstantly the linearly polarized wave in order to obtain the effect ofreducing the error correctly by the relationship of the formula (A).

[0251] If the incident light is not the linearly polarized wave, therelationship of the formula (A) is corrected accordingly, and a newresult is obtained, in which the following relationship minimizes theerror:

ΔβΔL=(π+δ)+2nπ[radian] (n: integer)  (A)′

[0252] where δ is the displacement for correction.

[0253] As described above, it is understood from the analysis in FIGS.3A to 6, that the error can be considerably reduced if ΔL is set tosatisfy the relationship of the formula (A) (or of the formula (A)′).

[0254] (Control of ΔβΔL)

[0255] From the above results, concerning the error generated by thepolarization dependency loss, the error is not dependent on ΔβL butdependent on only ΔβΔL if the angle of rotation at the splice point isclose to 90°. Making use of this nature, it is possible to maintain therelationship of the formula (A) irrespective of the change in ΔβL bycontrolling the sensing loop.

[0256] Further, as described above, the deviation fromΔβΔL=π+2nπ[radian] (n: integer) can be generated in the relationship ofthe formula (A) due to the fluctuation in the polarized condition of theincident light and other factors not included in the model. Even in sucha case, the error can be corrected to be minimum irrespective of thechange in ΔβL by controlling the length of ΔL.

[0257] This is a method of controlling ΔL so that the error is minimizedby the feedback control as described below.

[0258] For example, the magnitude of the error is dependent on the depthof modulation of the bias modulation to detect the resonance point,i.e., Δξ_(CW) and Δξ_(CCW) in the formulae (47) and (48).

[0259]FIG. 5E shows an example of comparison between a case in which thedepth of modulation set to be half width at half maximum at theresonance point is used (Δξ) and a case in which the depth of modulationset to one half (Δξ/2) is used when the error to ΔβΔL is ΔβL=2mπ (m:integer). FIG. 5F shows the comparison taking into consideration thesign of these errors.

[0260]FIG. 5F shows that, when the relationship of the formula (A) (orthe formula (A)′) is satisfied, the error is minimized irrespective ofthe depth of modulation, and for example, if two different depths ofmodulation are alternately switched at a predetermined frequency andapplied, the signal to indicate the deviation from the value to minimizethe error of ΔβΔL appears in the gyro output at the switching frequency.

[0261]FIG. 21 shows the conceptual view of eliminating the error by thismethod. The bias modulation to the CW light and the bias modulation tothe CCW light are achieved by two kinds of the depth of modulation, andapplied alternately at the predetermined switching frequency (f_(SW))(which is lower than the bias modulation frequency). The gyro outputwhen the depth of modulation 1 is applied and the gyro output when thedepth of modulation 2 is applied are different from each other in thegenerated error, and the error signal showing the deviation from theoptimum value of ΔL is generated at the switching period of the depth ofmodulation. This means that the error signal for the control is given.Using this error signal, ΔL is fed-back to control ΔL to an optimumvalue so that the error of the gyro output is minimized.

EXAMPLE

[0262] The example of the present invention is described based on theabove result of the analysis.

[0263]FIG. 7 shows a first embodiment of the resonator fiber optic gyrowith the method in accordance with the present invention appliedthereto, and the reflector ring resonator has the 90° splice.

[0264] When measuring the non-reciprocal effect of rotation, etc., ΔL isset to satisfy the relationship between the difference ΔL between thelengths L₁ and L₂ of two portions of the waveguide divided at apolarization-rotating point 9a in the coupler 8 and the ring resonatorand the difference Δβ in the propagation constant between twopolarization axes of the waveguide using the nature that the differenceinduced by the polarization fluctuation is minimized if ΔβΔL=π+2nπ[radian] (n: integer), or close thereto. The error induced by thepolarization fluctuation can be reduced thereby.

[0265]FIG. 8 shows a second embodiment of the resonator fiber optic gyrowith the method in accordance with the present invention appliedthereto, and the transmitter ring resonator has the 90° splice.

[0266] When measuring the non-reciprocal effect of rotation, etc., ΔLand L₃ are set to satisfy the relationship between the difference ΔLbetween the lengths L₁ from the input side coupler 13 to apolarization-rotating point 14 a, and the length (L₂+L₃) from thepolarization-rotating point 14 a to the input side coupler 13 via theoutput side coupler 15, the length L₃ between the input side coupler 13and the output side coupler 15, and the difference Δβ in the propagationconstant between two polarization axes of the waveguide using the naturethat the difference induced by the polarization fluctuation is minimizedif ΔβΔL=π+2nπ [radian] (n: integer), or close thereto, and ΔβL₃ =mπ[radian] (m: integer), or close thereto, where L₁, L₂ and L₃ are thelengths of three portions of the waveguide to be divided by the inputside coupler 13 (the first coupler) in which the light wave emitted fromthe laser beam source reaches first, the polarization-rotating point 14a in the ring resonator, and the output side coupler (the secondcoupler). The error induced by the polarization fluctuation can bereduced thereby.

[0267] The embodiments of the waveguide of the optical fiber aredescribed above, the present invention is, of course, applicable to anywaveguide other than those of the optical fiber.

[0268]FIG. 22 shows a third embodiment of the resonator fiber optic gyrowith the J method in accordance with the present invention appliedthereto. The configuration of the R-FOG using the reflector ringresonator having the 90° splice is shown in the figure.

[0269] In measuring the non-reciprocal effect of the rotation, etc., afiber is coiled around a columnar PZT (piezoelectric element) to controlthe length so that the difference ΔL between the lengths L₁ and L₂ oftwo portions of the waveguide (the optical fiber) divided by the coupler(Cl) and the 90° splice point in the ring resonator is optimized toreduce the error.

[0270] The following parts are added to the signal processing in thegeneral configuration (FIG. 12).

[0271] Two kinds of depth of modulation can be generated in both thebias modulation to the CW light and the bias modulation to the CCWlight, the switching signal of the frequency f_(SW) is generated by adepth of modulation switching period generator to switch the depth ofmodulation in a depth of modulation switching unit.

[0272] In the present embodiment, most of the processing are implementedthrough the digital logic; however, it matters little in the meaning ofthe principle whether the processing is implemented in an analog manneror in a digital manner. In addition, regarding the bias modulation, thebinary frequency shift through the “digital serrodyne modulation” inLiterature 1 may be implemented at a predetermined changing frequency.

[0273] The serrodyne wave in the figure is used to realize the closedloop, and the means is not limited thereto. In this embodiment, thedigitally generated serrodyne wave and the bias modulated wave areadded, and inputted in an electro-optic modulator on a LiNbO₃ waveguidetip (gyro tip). The step-like digital serrodyne wave (digital phaseramp) in which the time for the light wave to make one turn around theresonator is the width of the unit step as introduced in Literature 1,may be inputted. In this condition, the heights (φ1 and φ2) of one stepcorrespond to the frequencies (f₁ and f₂) of the serrodyne wave, and theangular velocity output (the gyro output) can be obtained from thedifference therebetween.

[0274] The angular velocity output includes the error signal showing thedeviation from the optimum value of ΔL at the depth of modulationswitching frequency f_(SW), and the error component is detected byextracting the error signal by a filter, and implementing thesynchronous detection by LIA3. The voltage is applied to the PZT so thatthe error component becomes zero, the fiber length is controlled, andthe feedback control is implemented so that ΔL is optimized.

[0275]FIG. 23 shows a forth embodiment of the resonator fiber optic gyrowith the method in accordance with the present invention appliedthereto. The configuration of the R-FOG using the transmitter ringresonator having the 90° splice is shown in the figure.

[0276] In measuring the non-reciprocal effect of the rotation, etc., afiber is coiled around a columnar PZT (piezoelectric element) to controlthe length so that the difference ΔL between the lengths L₁ from theinput side coupler (C1) to the 90° splice point, and the length (L₂+L₃)from the 90° splice point to the input side coupler (C1) via the outputside coupler (C2) is optimized to reduce the error, where L₁, L₂ and L3are the lengths of three portions of the waveguide (the optical fiber)to be divided by the input side coupler (C1) (the first coupler) inwhich the light wave emitted from the laser beam source reaches first,the 90° splice point in the ring resonator, and the output side coupler(C2) (the second coupler) to emit the light wave and input it in thelight receiver.

[0277] The signal processing method is similar to that when using thereflector ring resonator (FIG. 22).

[0278] As described above, in the present invention, the ring resonatoritself can be set in a condition in which no error is generated evenwhen the polarization dependency loss is present in the ring resonator.

[0279] Thus, the resonator fiber optic gyro using the method inaccordance with the present invention can minimize the error in the gyrooutput induced by the polarization fluctuation, and reduce the drift ofthe gyro output caused by the error fluctuation. A gyro with highaccuracy can be realized thereby.

What is claimed is:
 1. A method of reducing the polarization fluctuationinducing drift in a resonator fiber optic gyro, comprising the step of:setting ΔL so that the relationship of ΔL and Δβ satisfies a formulaΔβΔL=π+2nπ [radian] (n: integer), or approximately satisfies the formulato minimize the error induced by the polarization fluctuation where ΔLis the difference in length between L₁ and L₂ of two portions of awaveguide divided by a coupler and the polarization-rotating point, andΔβ is the difference in propagation constant of two axes of polarizationof the waveguide in a method of measuring the non-reciprocal effect suchas the rotation in a reflector ring resonator comprising a sensing loopformed of the waveguide having two axes of polarization to propagate thelight wave, and the coupler which is inserted in said sensing loop,guides the light wave from a laser beam source to said sensing loop andemits the light wave in said sensing loop, and having apolarization-rotating point in said sensing loop.
 2. A method ofreducing the polarization fluctuation inducing drift in a resonatorfiber optic gyro, comprising the step of: setting ΔL and L₃ so that therelationship of ΔL, Δβ and L₃ satisfies a formula ΔβΔL=π+2nπ [radian](n: integer) and ΔβL₃=mπ [radian] (m: integer), or approximatelysatisfies the formulae to minimize the error induced by the polarizationfluctuation where L₁ is the distance from said first coupler to thepolarization-rotating point, L₂ is the distance from thepolarization-rotating point to said second coupler, L₃ is the distancefrom said second coupler to said first coupler, ΔL is the differencebetween L₁ and the length (L₂+L₃) from the polarization-rotating pointto said first coupler through said second coupler, and Δβ is thedifference in propagation constant of two axes of polarization of thewaveguide when the waveguide is divided into three portions by saidfirst coupler, said polarization-rotating point and said second couplerin a method of measuring the non-reciprocal effect such as the rotationin a transmitter ring resonator comprising a sensing loop formed of thewaveguide having two axes of polarization to propagate the light wave, afirst coupler to guide the light wave from a laser beam source to saidsensing loop and a second coupler to emit the light wave in said sensingloop which are inserted in said sensing loop, and having apolarization-rotating point in said sensing loop.
 3. A method ofreducing the polarization fluctuation inducing drift in a resonatorfiber optic gyro as claimed in claim 1, wherein a polarization-rotatingangle at a polarization-rotating point is approximately 90°.
 4. A methodof reducing the polarization fluctuation inducing drift in a resonatorfiber optic gyro, comprising the step of: minimizing the errorirrespective of any change in ΔβL by controlling ΔL making use of thefact that the characteristic of said measurement error is considerablydependent on the product of Δβ and ΔL where ΔL is the difference inlength of two portions of a waveguide divided by said first coupler andsaid polarization-rotating point, and Δβ is the difference inpropagation constant of two axes of polarization of the waveguide, andless dependent on ΔβL which is the product of the sum of the length oftwo portions of the waveguide (L) and Δβ which is the sum in propagationconstant of two axes of polarization of the waveguide in a method ofmeasuring the non-reciprocal effect such as the rotation in a reflectorring resonator or a transmitter ring resonator comprising a sensing loopformed of the waveguide having two axes of polarization to propagate thelight wave and a coupler inserted in said sensing loop, and having apolarization-rotating point in said sensing loop.
 5. A method ofreducing the polarization fluctuation inducing drift in a resonatorfiber optic gyro as claimed in claim 4, wherein the error is minimizedirrespective of any change in ΔβL through the feedback to the differenceΔL in length between two portions of a waveguide divided by a coupler inwhich the light wave emitted from a laser beam source reaches first andthe polarization-rotating point in a ring resonator making use ofgeneration of an error signal indicating the deviation from an optimumvalue of ΔβΔL at a predetermined period by alternately applying twodifferent depths of modulation in the bias modulation implemented fordetecting a resonance point at the predetermined period.